3.2.63 \(\int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx\) [163]

Optimal. Leaf size=70 \[ -\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \]

[Out]

-1/2*(a-a*cos(x))^(1/2)/x^2-1/4*cot(1/2*x)*(a-a*cos(x))^(1/2)/x-1/8*csc(1/2*x)*Si(1/2*x)*(a-a*cos(x))^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3400, 3378, 3380} \begin {gather*} -\frac {1}{8} \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Cos[x]]/x^3,x]

[Out]

-1/2*Sqrt[a - a*Cos[x]]/x^2 - (Sqrt[a - a*Cos[x]]*Cot[x/2])/(4*x) - (Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x
/2])/8

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(2*a)^IntPart[n]
*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])), Int[(c + d*x)^m*Sin[e/2
+ a*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps

\begin {align*} \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx &=\left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x^3} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}+\frac {1}{4} \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {a-a \cos (x)} \left (4+2 x \cot \left (\frac {x}{2}\right )+x^2 \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right )\right )}{8 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cos[x]]/x^3,x]

[Out]

-1/8*(Sqrt[a - a*Cos[x]]*(4 + 2*x*Cot[x/2] + x^2*Csc[x/2]*SinIntegral[x/2]))/x^2

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a -a \cos \left (x \right )}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*cos(x))^(1/2)/x^3,x)

[Out]

int((a-a*cos(x))^(1/2)/x^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(x))^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(-a*cos(x) + a)/x^3, x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(x))^(1/2)/x^3,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(x))**(1/2)/x**3,x)

[Out]

Integral(sqrt(-a*(cos(x) - 1))/x**3, x)

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Giac [A]
time = 0.48, size = 48, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {2} {\left (x^{2} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) + 2 \, x \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) + 4 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a}}{8 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(x))^(1/2)/x^3,x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(x^2*sgn(sin(1/2*x))*sin_integral(1/2*x) + 2*x*cos(1/2*x)*sgn(sin(1/2*x)) + 4*sgn(sin(1/2*x))*sin
(1/2*x))*sqrt(a)/x^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a-a\,\cos \left (x\right )}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*cos(x))^(1/2)/x^3,x)

[Out]

int((a - a*cos(x))^(1/2)/x^3, x)

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